Let , or in general finite dimensional -vector space. Then we define (or ) as the set of all -dimensional subspaces (which are referred to as “-planes”) of . (Of course, we assume .)

Any -plane may be represented by an matrix of rank . That is, the rows of the matrix are basis elements of the -plane . Denote the set of all matrices with rank . Then we may identify

, the orbit of group action by .

**Exercise 1.1. **Justify the above identification by showing the following statement. Any matrices and represent same -planes if and only if there exists such that .

(Hint: Show that represent the same -plane if and only if there is matrix such that for each , where denote the rows of , respectively. Let and denote the th columns of and respectively. Take the transpose to get . Thus represent the same -planes if and only if there are matrices such that and . Show that so that .)

**Exercise 1.2.** Show that .

(Hint: From the hint given for Exercise 1.1., show that for any matrices , the row spaces of coincide if and only if the column spaces of them coincide. Then for any rank matrix , we may assume that is given by augmenting an matrix on the right to a invertible matrix (i.e., an element of ). Argue that the entries of decide the dimension of the whole grassmannian.)

Recall that we can projectivize a vector space as follows:

.

If then clearly . We have the identification .

**Example.** We have

.

There is another way to describe . We review the relevant linear algebra first.

Recall that is the set of alternating -multilinear functions . Given and , we define

.

That is, we have

.

**Exercise 1.3.** Show that .

**Exercise 1.4.** Let be the standard dual basis. Given and , show that , which is when and otherwise. Conclude that the set form a basis for . (In particular, we have , and of course we are assuming . If , exterior -power is just zero.)

**Convention**. Given , we have a unique dual image . Thus, if , then we write .

**Exercise 1.5.** For the standard basis , show that .

**Exercise 1.6.** Given the bases , take the matrix given by . In other words, in basis the th row of are . Show that

.

Thus if is an matrix, then

.

Now we describe in a different way. Fix any . Consider any basis . Then we have a map . If we write and , then is the matrix whose rows are . Writing , we have . Thus for any other basis , we have

.

Now define by choosing a basis since is not dependent on the choice of basis of .

More concretely, if we write . Then , where . This means that we can recognize as the space of skew-symmetric matrices:

.

Since , we have

.

The following exercise gives an intuitive picture of .

**Exercise 1.7. **Given , define . Given , show .

**Exercise 1.8.** Given an skew symmetric matrix , if is odd, then .

**Exercise 1.9. (Hard)** Given an skew symmetric matrix , if is even and , then there is a polynomial such that . This polynomial is called the **Pfaffian** of .

**Exercise 1.10.** Explain how to define the rank of elements in (Hint: use skew symmetric matrix).